Verify the following identities sin 3 theta 3 sin theta 4

Verify the following identities: sin 3 theta = 3 sin theta - 4 sin3 theta

Hint: write 3 theta = 2 theta + theta and use the sum formula followed by an application of the double angle identities and finally the Pythagorean identity.

Solution

prove sin3@ = 3*sin@ - 4*sin³@

LHS:

sin3@ = sin( 2@ + @)

= sin2@ *cos@ + cos2@ *sin@ [ sin(A+B) = sinA cosB + cosA sinB]

= ( 2*sin@*cos@) *cos@ + ( 1-2*sin²@)*sin@

= 2*sin@*cos²@ + sin@ - 2*sin³@

= 2*sin@ ( 1-sin²@ ) + sin@ - 2*sin³@ [ sin²@ + cos²@ = 1 , cos²@ = 1-sin²@]

= 3sin@ - 4*sin³@

= RHS

LHS = RHS